{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "nbsphinx": "hidden"
   },
   "source": [
    "# Characterization of Systems in the Spectral Domain\n",
    "\n",
    "*This Jupyter notebook is part of a [collection of notebooks](../index.ipynb) in the bachelors module Signals and Systems, Communications Engineering, Universität Rostock. Please direct questions and suggestions to [Sascha.Spors@uni-rostock.de](mailto:Sascha.Spors@uni-rostock.de).*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Phase and Group Delay\n",
    "\n",
    "The [phase and group delay](https://en.wikipedia.org/wiki/Group_delay_and_phase_delay) characterize the phase and delay properties of an LTI system. Both quantify the frequency dependent delay that is imprinted on a signal when passing through a system. In many applications the delay introduced by a system should be as small as possible or within reasonable limits."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Phase Delay\n",
    "\n",
    "For an LTI system with transfer function $H(j \\omega)$ the phase delay is defined as follows\n",
    "\n",
    "\\begin{equation}\n",
    "t_p(\\omega) = - \\frac{\\varphi(j \\omega)}{\\omega}\n",
    "\\end{equation}\n",
    "\n",
    "where $\\varphi(j \\omega) = \\arg \\{ H(j \\omega) \\}$ denotes the phase of the transfer function. The phase delay quantifies the delay of a single harmonic exponential signal $e^{j \\omega t}$ with frequency $\\omega$ when passing through the system. The negative sign in the definition of the phase delay results in a positive phase delay $t_p(\\omega) > 0$ when a signal is delayed by a system. Note that the phase delay may not be defined for $\\omega = 0$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Example**\n",
    "\n",
    "As example, the phase delay $t_p(\\omega)$ is computed for the [2nd order low-pass filter introduced before](../laplace_transform/network_analysis.ipynb#Example:-Second-Order-Low-Pass-Filter). First the transfer function $H(j \\omega)$ is defined in `SymPy`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{1}{- C L \\omega^{2} + i C R \\omega + 1}$"
      ],
      "text/plain": [
       "          1           \n",
       "──────────────────────\n",
       "       2              \n",
       "- C⋅L⋅ω  + ⅈ⋅C⋅R⋅ω + 1"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import sympy as sym\n",
    "sym.init_printing()\n",
    "\n",
    "L, R, C = sym.symbols('L R C', positive=True)\n",
    "w = sym.symbols('omega', real=True)\n",
    "s = sym.I * w\n",
    "\n",
    "H = 1 / (C*L*s**2 + C*R*s + 1)\n",
    "H"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now the phase delay $t_p(\\omega)$ is computed"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - \\frac{\\arg{\\left(\\frac{1}{- C L \\omega^{2} + i C R \\omega + 1} \\right)}}{\\omega}$"
      ],
      "text/plain": [
       "    ⎛          1           ⎞ \n",
       "-arg⎜──────────────────────⎟ \n",
       "    ⎜       2              ⎟ \n",
       "    ⎝- C⋅L⋅ω  + ⅈ⋅C⋅R⋅ω + 1⎠ \n",
       "─────────────────────────────\n",
       "              ω              "
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "phi = sym.arg(H)\n",
    "tp = - phi/w\n",
    "tp"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and the result is visualized using the normalized values $R=1$, $L=0.5$ and $C=0.4$ for the elements of the low-pass filter"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/pdf": 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      "text/plain": [
       "<Figure size 614.4x460.8 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x106658208>"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "RLC = {R: 1, L: sym.Rational('.5'), C: sym.Rational('.4')}\n",
    "sym.plot(tp.subs(RLC), (w, -10, 10),\n",
    "         xlabel='$\\omega$', ylabel='$t_p(j \\omega)$')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Group Delay\n",
    "\n",
    "The group delay is defined as the derivative of the phase with respect to the frequency\n",
    "\n",
    "\\begin{equation}\n",
    "t_g(\\omega) = - \\frac{d \\varphi(j \\omega)}{d \\omega}\n",
    "\\end{equation}\n",
    "\n",
    "The group delay quantifies the delay the amplitude envelope of a group of exponential signals observes when passing through a system. The negative sign in above definition results in a positive group delay for a system imposing a delay onto the input signal. Note that the [phase](https://en.wikipedia.org/wiki/Instantaneous_phase) $\\varphi(j \\omega)$ is in general only unique for $- \\pi < \\varphi(j \\omega) \\leq \\pi$. If the phase exceeds this range it is wrapped back. For meaningful results it is required to unwrap the phase before computing the group delay."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Example**\n",
    "\n",
    "The group delay $t_g(\\omega)$ of above 2nd order low-pass filter is computed and plotted for the normalized values"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
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\n",
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   "source": [
    "tg = - sym.diff(phi, w)\n",
    "sym.plot(tg.subs(RLC), (w, -10, 10),\n",
    "         xlabel='$\\omega$', ylabel='$t_g(j \\omega)$')"
   ]
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   "source": [
    "**Copyright**\n",
    "\n",
    "This notebook is provided as [Open Educational Resource](https://en.wikipedia.org/wiki/Open_educational_resources). Feel free to use the notebook for your own purposes. The text is licensed under [Creative Commons Attribution 4.0](https://creativecommons.org/licenses/by/4.0/), the code of the IPython examples under the [MIT license](https://opensource.org/licenses/MIT). Please attribute the work as follows: *Sascha Spors, Continuous- and Discrete-Time Signals and Systems - Theory and Computational Examples*."
   ]
  }
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